ELLIPTIC CURVES, L-FUNCTIONS, AND HILBERT’S TENTH PROBLEM
M. RAM MURTY AND HECTOR PASTEN
To the memory of Anthony V. Geramita.
Abstract. Hilbert’s tenth problem for rings of integers of number fields remains open in general,
although a negative solution has been obtained by Mazur and Rubin conditional to a conjecture on
Shafarevich-Tate groups. In this work we consider the problem from the point of view of analytic
aspects of L-functions instead. We show that Hilbert’s tenth problem for rings of integers of number
fields is unsolvable, conditional to the following conjectures for L-functions of elliptic curves: the
automorphy conjecture and the rank part of the Birch and Swinnerton-Dyer conjecture.
1. Introduction
In 1900, Hilbert asked for an algorithm to decide the Diophantine problem of Z. Namely, he asked
for an algorithm which takes as input a polynomial equation with integer coefficients (possibly in
many variables) and after a finite amount of computation can determine if the equation has an integer
solution or not. This problem is known as Hilbert’s tenth problem.
Matijasevich proved in 1970 that the Diophantine problem of Z is undecidable [16], after the work
of Davis, Putnam, and Robinson [4] thus showing that Hilbert’s tenth problem is unsolvable. It is
natural to address the analogous question in other rings with interesting arithmetic, such as number
fields, meromorphic functions, function fields, their rings of integers, etc. (See Section 2 for a precise
statement.) In this paper we consider the case of rings of integers of number fields.
Let K be a number field and let OK be its ring of integers. The current approach to showing that
the Diophantine problem for OK is undecidable consists of trying to show that Z has a Diophantine
definition in OK , thus reducing the problem to Matijasevich’s theorem. This approach has been
successful in the following cases:
• K is totally real, or a quadratic extension of a totally real number field [6, 8, 7],
• K has exactly one non-real archimedean place [23, 28],
• subfields of the previous ones (e.g. all abelian extensions of Q) [26]
By work of Poonen [24], Cornelissen-Pheidas-Zahidi [3], and Shlapentokh [30] the sought Diophantine definition of Z in OK can be obtained from the existence of suitable elliptic curves over number
fields retaining their rank in finite extensions. This important link with the theory of elliptic curves
was used by Mazur and Rubin [18] to prove the following
Theorem 1.1 (Mazur-Rubin). Assume the squareness conjecture for the 2-torsion part of ShafarevichTate groups of elliptic curves over number fields. Then for every number field K, the Diophantine
problem of OK is undecidable.
The conjecture assumed in this result says that, if K is a number field and E is an elliptic curve over
K, then the finite 2-group XE [2] has square order. This would follow from the folklore conjecture that
XE is finite, which in turn is implied by the special value formula in the Birch and Swinnerton-Dyer
conjecture.
Date: July 20, 2016.
2010 Mathematics Subject Classification. Primary 11U05; Secondary 11G05, 14G10 .
Key words and phrases. Hilbert’s tenth problem, rings of integers, ranks of elliptic curves.
M. R. M. was partially supported by an NSERC Discovery grant. H. P. was partially supported by a Benjamin
Peirce Fellowship at Harvard, and by a Schmidt Fellowship and the NSF agreement No. DMS-1128155 at the IAS. Any
opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not
necessarily reflect the views of the National Science Foundation.
1
2
M. RAM MURTY AND HECTOR PASTEN
In this paper we address the existence of the necessary elliptic curves from the point of view of
L-functions. Our goal is to show that if L-functions of elliptic curves have the “expected good analytic
properties” and if they satisfy the rank part of the Birch and Swinnerton-Dyer conjecture, then Hilbert’s
tenth problem for OK is unsolvable for every number field K.
Our main results are presented in sections 6 and 7 below, in particular see Theorem 6.2 and Theorem
7.1. An immediate consequence which requires less preparation (although it is somewhat weaker) is
the following
Theorem 1.2. Suppose that elliptic curves over number fields are automorphic, that they satisfy the
parity conjecture, and that they satisfy the analytic rank zero part of the twisted Birch and SwinnertonDyer conjecture. Then for every number field K, Hilbert’s tenth problem for OK is unsolvable (i.e. the
Diophantine problem for OK is undecidable).
Let us explain the conjectural hypotheses in this result. Let K be a number field. An elliptic
curve E over K is automorphic if its L-function L(s, E) agrees (up to some normalization) with the
L-function of a cuspidal automorphic representation of GL2 , and it is conjectured that this is always
the case [13]. This is a generalization to number fields of the Shimura-Taniyama conjecture (now the
modularity theorem), see Section 3 for more details. By the Jacquet-Langlands converse theorem for
GL2 (see [14]), this is the same as requiring that the suitably defined L-functions L(s, E, η) twisted
by certain Hecke characters η, have “good analytic properties” including analytic continuation and
functional equation.
The parity conjecture asserts that the global root number of E/K equals (−1)rk E(K) . For automorphic elliptic curves the root number of E equals the sign of the functional equation of L(s, E), so
the parity conjecture follows from the rank part of the Birch and Swinnerton-Dyer (BSD) conjecture,
which asserts that
ords=1 L(s, E) = rk E(K).
On the other hand, by the “twisted Birch and Swinnerton-Dyer conjecture” we mean the variation of the
BSD conjecture involving twists by finite order Hecke characters, as proposed by Mazur [17]. Namely,
if L/K is a finite abelian extension and η is a finite order Hecke character corresponding to it, then
ords=1 L(s, E, η) equals the dimension of the η-isotypical component of the Gal(L/K)-representation
E(L) ⊗ C. Note that when η is the trivial character one recovers the rank part of the BSD conjecture.
The analytic rank zero part mentioned in Theorem 1.2 refers to the conjectural implication
ords=1 L(s, E, η) = 0 =⇒ dim(E(L) ⊗ C)η = 0.
Actually, Theorem 1.2 needs milder hypotheses (see Theorem 6.2), some of which are known to hold
in certain degree of generality thanks to the current spectacular progress on the Langlands program
and on the Birch and Swinnerton-Dyer conjecture made by several authors. This will be discussed in
Section 7.
Let us briefly explain the connection between Hilbert’s tenth problem and L-functions. By the work
of Poonen and Shlapentokh, one only needs to show that for cyclic number field extensions L/K of
prime degree, there is an elliptic curve E defined over K such that rk E(L) = rk E(K) = 1 (in fact, this
is precisely what is obtained by Mazur and Rubin under the squareness conjecture for XE [2]). A priori,
if one assumes automorphy and BSD in order to address the problem in the analytic side, one ends up
trying to control simultaneously the vanishing order of two L-functions for automorphic representations
of GL2 , a problem which is currently out of reach (see [2] for a discussion on this problem in a general
theoretical framework; the discussion in p. 169 is particularly relevant here). We can circumvent
this difficulty thanks to a result of Shlapentokh which shows that actually rk E(L) = rk E(K) > 0
suffices for the undecidability application. The analytic counterpart of this last condition can be
translated into the problem of controlling just one L-function for an automorphic representation of
GL2 varying over quadratic twists, under some additional congruence restrictions for the admissible
quadratic twists. The necessary non-vanishing results are provided by theorems of Friedberg-Hoffstein
[12], see also Murty-Murty [19] and [20] for the case K = Q. It is crucial that these non-vanishing
results are also applicable to non-self-contragredient automorphic representations (in the case K = Q,
this corresponds to modular forms with non-trivial nebentypus).
HILBERT’S TENTH PROBLEM FOR RINGS OF INTEGERS
3
We remark that the elliptic curves retaining their positive rank in number field extensions produced
(conditionally) in our work as well as in [18], can also be used to obtain definability and decidability
consequences in the context of Hilbert’s tenth problem for big sub-rings of number fields (i.e. rings of
S-integers of number fields with S a set of primes with positive natural density). We refer the reader
to Theorem 1.9 in [30] for details.
We conclude this introduction with an outline of the paper. The necessary background on Hilbert’s
tenth problem, automorphic L-functions, and the BSD conjecture is given in sections 2, 3, and 4
respectively. Our results will only apply to elliptic curves satisfying certain conditions on their global
root numbers, so in Section 5 we produce elliptic curves with the necessary properties. The main
results in the context of Hilbert’s tenth problem are given in Section 6. Finally, in Section 7 we discuss
some arithmetic applications and unconditional results for L-functions, mainly related to elliptic curves
retaining their positive rank in cyclic extensions of totally real number fields.
2. Hilbert’s tenth problem
Let R be a (commutative, unitary) ring, and let R0 be a recursive sub-ring of R. Then the polynomial
ring in countably many variables R0 [x1 , x2 , ...] is also recursive, and one can formulate the following
analogue of Hilbert’s tenth problem:
Problem 2.1 (H10(R; R0 )). Find an algorithm (in the sense of a Turing machine) that takes as input
polynomial equations (possibly with many variables) with coefficients in R0 , and decides whether the
equation has a solution over R or not.
In other words, H10(R; R0 ) asks for an algorithm1 to solve the Diophantine problem of R with
coefficients in R0 . When the requested algorithm exists, we say that H10(R; R0 ) is decidable, otherwise
we say that it is undecidable.
Let us stress the fact that H10(R; R0 ) is sensitive not only to R but also to R0 . For instance,
H10(C[z]; Z) is decidable (the problem can be reduced to H10(C; Z)) but H10(C[z]; Z[z]) is undecidable
by [5].
We will be interested in the case when K is a number field and R = R0 = OK is the ring of integers
of K. So, instead of continuing with a general discussion, let us focus on this particular case and write
H10(OK ) instead of H10(OK ; OK ). Note that the case K = Q is precisely Hilbert’s original problem
and it is known to be undecidable by work of Davis, Putnam, Robinson and finally Matijasevich.
It is widely conjectured that H10(OK ) is undecidable for every number field K, and the general
approach for showing results in this direction consists of showing that Z is Diophantine in OK , in the
n
is Diophantine if there is a polynomial F ∈ OK [x1 , ...xn , y1 , ..., ym ] such
following sense: a set S ⊆ OK
that
n
m
S = {a ∈ OK
: ∃b ∈ OK
, F (a, b) = 0}.
In fact, it is easy to see that if Z is Diophantine in OK then an algorithm as the one requested in
H10(OK ) can be modified to get an algorithm for H10(Z), which is not possible by Matijasevich’s
theorem. We remark, however, that for the purpose of showing that H10(OK ) is undecidable, it would
suffice to give a Diophantine interpretation of Z rather that an actual Diophantine definition.
The previous approach has had remarkable success, and now one knows that Z is Diophantine in
OK in the following cases (cf. the introduction):
• K is contained in a CM number field (imaginary quadratic extension of totally real);
• K is contained in a number field with exactly one non-real archimedean place.
More generally, we say that a number field extension L/K is integrally Diophantine if OK is Diophantine in OL . This property of number field extensions enjoys some very useful “functorial” properties.
Proposition 2.2. The following holds for number field extensions:
• If L/K is integrally Diophantine and H10(OK ) is undecidable, then so is H10(OL );
• If L/K and K/k are integrally Diophantine, then so is L/k;
1One can argue that a better question is to ask whether the Diophantine problem is decidable or not. We preferred
to keep Hilbert’s formulation for the sake of historical accuracy.
4
M. RAM MURTY AND HECTOR PASTEN
• If L/k is integrally Diophantine and K is an intermediate field, then K/k is integrally Diophantine;
• If L/K1 and L/K2 are integrally Diophantine, then so is L/K1 ∩ K2 .
See [8] and [26] for details, and see Chapter 2 of [29] for a general reference on this subject.
For instance, from the previous proposition it follows that H10(OK ) is undecidable whenever K/Q
is abelian: in that case K is contained in a cyclotomic field by the Kronecker-Weber theorem, and
cyclotomic fields are CM.
The following result is due to Poonen and Shlapentokh. It reduces the problem of verifying if an
arbitrary Galois extension of number fields is integrally Diophantine to a much simpler class of Galois
extensions. The proof is the same as in Corollary 8.4 of [18]; we include a proof here for the convenience
of the reader.
Proposition 2.3. Let L/K be a Galois extension of number fields. Suppose that every cyclic extension
of prime degree k 0 /k with K ⊆ k ⊆ k 0 ⊆ L satisfies that k 0 /k is integrally Diophantine. Then L/K is
integrally Diophantine.
In particular, if every cyclic extension of prime degree of number fields is integrally Diophantine,
then for every number field K one has that Z is Diophantine in OK and H10(OK ) is undecidable.
Proof. Let k be any intermediate field of K/L such that L/k is cyclic, and consider a tower of fields
L = kn ⊇ ... ⊇ k1 ⊇ k0 = k where each ki /ki−1 is cyclic of prime degree. Then by hypothesis each
ki /ki−1 is integrally Diophantine, thus L/k is integrally Diophantine.
Let G = Gal(L/K). For each g ∈ G we get that L/Lhgi is integrally Diophantine, therefore L/K is
integrally Diophantine because K = LG = ∩g∈G Lhgi .
Let us now discuss the relation between integrally Diophantine number field extensions and elliptic
curves retaining their rank in extensions. The observation that elliptic curves preserving their ranks
in number field extensions K/k play a prominent role in Hilbert’s tenth problem for rings of integers,
can be traced back to the work of Denef [7]. At the end of that paper, and despite already having an
unconditional result for totally real number fields, Denef notes that if L is a totally real number field
and there is an elliptic curve E/Q with
rk E(L) = rk E(Q) > 0
then one can show Z is Diophantine in OL in a way simpler than in his unconditional proof.
In 2002, Poonen [24] gave related elliptic curve criterion (with a proof different to the argument of
Denef) and applicable in more generality:
Theorem 2.4 (Poonen). Let L/K be a number field extension and suppose that there is an elliptic
curve E/K such that
rk E(L) = rk E(K) = 1.
Then L/K is integrally Diophantine.
Specializing to the case K = Q and motivated by Hilbert’s tenth problem, Poonen asked [24] whether
it is true that for every L there is an elliptic curve E/Q with rk E(L) = rk E(Q) = 1. It was later
observed by Mazur and Rubin that this would contradict standard parity conjectures on elliptic curves.
Nevertheless, in order to show that K/Q is integrally Diophantine for every number field K it suffices
to verify Poonen’s elliptic curve criterion for cyclic extensions of prime degree, and this is precisely
what Mazur and Rubin achieve in [18] assuming the squareness conjecture for the 2-torsion part of
Shafarevich-Tate groups (which follows from the finiteness conjecture for Shafarevich-Tate groups).
Poonen’s criterion was generalized by Cornelissen-Pheidas-Zahidi [3] and later by Shlapentokh (and
independently by Poonen) [30] to relax the rank condition. Let us recall here Shlapentokh’s version:
Theorem 2.5 (Shlapentokh). Let L/K be a number field extension and suppose that there is an elliptic
curve E/K such that
rk E(L) = rk E(K) > 0.
Then L/K is integrally Diophantine.
HILBERT’S TENTH PROBLEM FOR RINGS OF INTEGERS
5
It is this last elliptic curve criterion what we will verify in cyclic extensions of prime degree, under
suitable conjectures regarding the L-functions of elliptic curves. In fact, it is very important for our
approach that positivity of the rank suffices, rather than the more restrictive requirement that the
rank be equal to 1.
3. Automorphic forms and L-functions
We will be interested in the study of L-functions of automorphic elliptic curves, for which we need
the language of automorphic representations. The standard reference for the GL2 theory (which is
all we need) is [14]. The article of Gelbart [13] on automorphic elliptic curves is also relevant to our
discussion. On the other hand, in this section we also recall the necessary non-vanishing results for
quadratic twists of automorphic L-functions.
Let K be a number field and let π be an irreducible cuspidal automorphic representation of
GL2 (AK ), where AK denotes the adele ring of K. Let L(s, π) be the (completed) L-function attached to π and the standard representation of GL2 . Then L(s, π) is entire and satisfies a functional
equation
L(s, π) = (s, π)L(1 − s, π̃)
where π̃ is the contragredient representation of π and (s, π) is the corresponding global epsilon factor.
Let η be an irreducible automorphic representation of GL1 (AK ); it is 1-dimensional by Tate’s theory.
The representation η can be lifted to an automorphic representation of G(AK ) by composition with the
determinant map, so that one obtains an irreducible automorphic representation π ⊗ η := π ⊗ (η ◦ det)
of GL2 (AK ). Let us remark that the contragredient representation is then (π ⊗ η)∼ = π̃ ⊗ η −1 , and
that the local factors of L(s, π ⊗ η) at all but finitely many places agree with the local factors of the
Rankin-Selberg L-function L(s, π × η).
Let S be a finite set of places of K and let Ψsp (S) be the set of (isomorphism classes of) irreducible
automorphic representation of GL1 (AK ) of order 2, unramified at every v ∈ S, with the property that
their kernels contain the uniformizer at v for each non-archimedean v ∈ S. The set Ψsp (S) is infinite
and it corresponds, under class field theory, to the set of quadratic extensions of K that are unramified
at every v ∈ S, and split at v if v ∈ S is non-archimedean.
The following non-vanishing results are due to Friedberg and Hoffstein [12]. The case K = Q is
also proved by completely different means by Murty and Murty [20] in the language of modular forms.
(The results actually hold for more general choices of congruence conditions than our Ψsp (S).)
Theorem 3.1. Let S be a finite set of places of K and let π be a non-self-contragredient cuspidal
automorphic representation of GL2 (AK ). There are infinitely many η ∈ Ψsp (S) such that
L(1/2, π ⊗ η) 6= 0.
Theorem 3.2. Let S be a finite set of places of K and let π be a self-contragredient cuspidal automorphic representation of GL2 (AK ). Suppose that there is some η ∈ Ψsp (S) such that (1/2, π ⊗ χ) = 1.
Then there are infinitely many ω ∈ Ψsp (S) such that
L(1/2, π ⊗ ω) 6= 0.
If E is an elliptic curve defined over K, we let L(E, s) be the completed L-function of E. The
automorphy conjecture for elliptic curves is the following (see Gelbart’s article [13] for a detailed study
of this conjecture):
Conjecture 3.3 (Automorphy conjecture). If E is an elliptic curve over K, then there is a cuspidal
automorphic representation πE of GL2 (AK ) satisfying
L(s, E) = L(s − 1/2, πE ).
Elliptic curves for which this conjecture holds are called automorphic. Remarkable progress on the
above conjecture has been made for elliptic curves over totally real number fields. The literature on
automorphy of elliptic curves in this last setting is constantly growing, but nevertheless, we refer the
reader to [11] for some of the most recent developments and for further references.
Let L/K be an abelian extension with Galois group G and let χ vary over the irreducible characters
of G. Then one can form the twisted L-functions L(s, E, χ) by modifying the local factors by χ(Frobp )
6
M. RAM MURTY AND HECTOR PASTEN
at unramified places (a more subtle definition is used at the remaining finitely many places). By
comparing local factors, it can be verified that the L-function of the base change of E to L satisfies
Y
L(s, EL ) =
L(s, E, χ)
χ
where for the trivial character 1, one has L(s, E, 1) = L(s, E).
In the particular case when [L : K] = 2 (so that there is only one non-trivial character χ = χL ) we
have another factorization L(s, EL ) = L(s, E)L(s, E L ) where E L is the elliptic curve over K defined
as the quadratic twist of E by L. Thus, in particular we find in this case that L(s, E, χL ) = L(s, E L ).
Now we return to the general case when L/K is abelian. If ηχ denotes the finite order irreducible
representation of GL1 (AK ) obtained from χ by class-field theory by composition with the Artin map,
and if E is automorphic, then it follows by a local computation that
L(s, E, χ) = L(s − 1/2, πE ⊗ ηχ )
which implies that each L(s, E, χ) (hence, L(s, EL )) extends to an entire function satisfying the appropriate functional equation relating s and 2 − s.
4. The Birch and Swinnerton-Dyer conjecture
Let K be a number field and E an elliptic curve defined over K. The L-function L(s, E), defined
as an Euler product, converges for <(s) > 3/2. One expects that L(s, E) can be extended to an entire
function, in which case it makes sense to consider the vanishing order of it at s = 1.
Suppose that E is automorphic, so that analytic continuation holds. The rank part2 of the Birch
and Swinnerton-Dyer conjecture (BSD) asserts that
rk E(K) = ords=1 L(s, E).
Moreover, the automorphic representation πE is self-contragredient so that the functional equation
takes the form
L(s, E) = (s, E)L(2 − s, E)
with (1, E) = (1/2, πE ) ∈ {−1, 1}, which is often referred to as the sign of the functional equation.
It is known that the sign of the functional equation is equal to w(E), the global root number of E over
K (which can be defined even when E is not automorphic). Then, in the case of automorphic elliptic
curves, the rank part of BSD implies the parity conjecture, which we now recall.
Conjecture 4.1 (Parity conjecture). Let K be a number field and let E be an elliptic curve over K.
Then the global root number satisfies w(E) = (−1)rk E(K) .
(Note that the parity conjecture makes sense even for elliptic curves that are not known to be
automorphic.)
Let L/K be a Galois extension of number fields with Galois group G, and let E be an elliptic curve
over K. A generalization of the rank part of the BSD, called the equivariant BSD conjecture, predicts
a relation between the dimensions of the isotypical components of the G-representation V (E, L) :=
E(L)⊗C and the vanishing order at 1 of certain L-functions twisted by irreducible representations. Let
us restrict our attention to the case when L/K is abelian, in which case the conjecture was proposed
by Mazur [17] and we will refer to it as the twisted BSD conjecture.
In the abelian case we have a factorization (cf. previous section)
Y
L(s, EL ) =
L(s, E, χ)
χ
where χ varies over the irreducible characters of G (a similar factorization holds without the abelian
hypothesis). If E is automorphic, then each L(s, E, χ) is automorphic as explained in the previous
2There is also a conjectural formula for the first non-zero term in the Taylor expansion –we will not need that part
of the BSD conjecture.
HILBERT’S TENTH PROBLEM FOR RINGS OF INTEGERS
7
section, and one can consider their vanishing orders at s = 1. On the other hand, we have an isotypical
decomposition
M
V (E, L) =
V (E, L)χ .
χ
Conjecture 4.2 (Twisted BSD conjecture). With the previous notation and assumptions, for each χ
we have
dim V (E, L)χ = ords=1 L(s, E, χ).
Note that for the trivial character χ = 1 one has V (E, L)1 = E(K) ⊗ C and L(s, E, 1) = L(s, E), so
that one recovers the original rank part of BSD. In general, however, it is not known if the rank part
of BSD implies the general abelian case of the equivariant BSD.
Conjecture 4.3 (Twisted analytic rank zero conjecture). Let K be a number field, let L/K be finite
abelian with Galois group G, and let E be an automorphic elliptic curve over K. Let χ be an irreducible
character of G. Then one has the following implication:
L(1, E, χ) 6= 0 =⇒ dim V (E, L)χ = 0.
The twisted analytic rank zero conjecture is certainly more accessible than the full equivariant BSD,
and in fact, some important cases are known unconditionally. See for instance [25], [15], [21] and the
references therein.
5. Root numbers
The results in the next section will apply to elliptic curves satisfying certain condition on global
root numbers. In this section we will give a supply of such elliptic curves.
First, let us recall the following from [9]:
Lemma 5.1. Let K be a number field and let E be an elliptic curve defined over K. If K has some
real place, or if E does not have potentially good reduction everywhere, then E has some quadratic
twist with global root number −1.
We will also need elliptic curves that acquire negative root number in a fixed quadratic extension.
Lemma 5.2. Let L/K be a quadratic extension of number fields. There are infinitely many elliptic
curves E defined over K such that w(E ⊗L) = −1 and having pairwise distinct j-invariants. Moreover,
these curves can be taken semi-stable or quadratic twist of semi-stable.
Proof. Let p - 2 be a prime of K inert in L/K and let PL be the prime of L above it. Let E be a
semi-stable elliptic curve defined over K with multiplicative reduction at p and with vp (jE ) < 0 (so
that E does not have potentially good reduction at p). Such an E exists and can be taken of the form
y 2 + y = x3 − x2 + t for suitable t ∈ OK , see the proof of Lemma 5.4 in [18]; the requirement on j
invariants can be achieved in this way. Let M/K be a quadratic extension which is ramified at p (call
PM the prime of M above p) and split at each of the following places: primes 6= p where E has bad
reduction, primes that ramify in L/K, and all places at infinity. (This initial setup for the proof is
inspired by the initial choices in the proof of Proposition 6.1 [18].) Let F = M L and note that it is a
quadratic extension of M and of L due to the ramification conditions at p. Moreover, F/L ramifies at
PL and we let PF be the only prime above PL , so that PF is the only prime above p in F/K.
We will show that either w(E ⊗ L) = −1 or w(E M ⊗ L) = −1. This will prove the lemma after
replacing E by E M if necessary.
We have
w(E ⊗ F ) = w(E ⊗ M )w((E ⊗ M )F ) = w(E ⊗ M )w(E L ⊗ M )
and by our splitting hypotheses on M/K, we see that local root numbers at primes not dividing p
cancel-out. Thus we get w(E ⊗ F ) = wPF (E ⊗ F ), the local root number at PF . Since E has
multiplicative (non-potentially good) reduction at p and since p is inert in L/K we see that E ⊗ L
has split multiplicative reduction at PL , hence E ⊗ F has split multiplicative reduction at PF and we
deduce wPF (E ⊗ F ) = −1. Finally, we obtain
−1 = w(E ⊗ F ) = w(E ⊗ L)w((E ⊗ L)F ) = w(E ⊗ L)w(E M ⊗ L).
8
M. RAM MURTY AND HECTOR PASTEN
We remark that the previous lemma does not hold for arbitrary number field extensions. In fact,
there are number field
L/K of degree 4 for which each elliptic curve E/K satisfies w(E⊗L) =
√
√ extensions
1. For instance, Q( −1, 17)/Q is such an extension. See Remark 7.7 in [18] and see also [9].
6. Elliptic curves retaining their rank
The next lemma is a simple remark, which nevertheless is central to our approach. In fact, it is
the technical reason for considering cyclic extensions of prime degree instead of more general abelian
extensions.
Lemma 6.1. Let V be a Q-vector space of finite dimension and let G → AutQ (V ) be a representation
of a finite group G of prime order p. All non-trivial irreducible representations of G appear with the
same multiplicity in VC := V ⊗ C.
Our main result is the following:
Theorem 6.2. Let L/K be a cyclic extensions of number fields with prime degree p, and let G be the
Galois group of L/K. Let E be an elliptic curve over K and suppose that it satisfies the following:
(i) If p = 2 then w(E ⊗ L) = −1, and if p > 2 then E has some quadratic twist with global root
number −1;
(ii) E is automorphic over K;
(iii) The quadratic twists of E satisfy the parity conjecture over K;
(iv) The quadratic twists of E satisfy the twisted analytic rank zero conjecture (over K) for the
non-trivial characters of G;
Then there are infinitely many quadratic extensions M/K for which the quadratic twist E M /K satisfies
that
rk E M (L) = rk E M (K) > 0.
Proof. Let E/K satisfy (i)-(iv). By (i), possibly after replacing E by a suitable quadratic twist we
can assume that w(E) = −1 (for p = 2 either E or E L works) while (i)-(iv) still hold for this new E.
Let S be a finite set of places of K containing all archimedean places and all places of bad reduction
of E. Let χ be a fixed non-trivial irreducible character of G. We claim that there are infinitely many
quadratic extensions M/K such that the quadratic twist E M satisfies
(a) ords=1 L(s, E M ) is odd, and
(b) L(1, E M , χ) 6= 0.
We need to consider separately the cases when p = 2 and p ≥ 3.
Case p = 2. By (i) we have w(E L ) = 1. The representation πE ⊗ ηχ = πE L is self-contragredient
and the sign of the functional equation is (1/2, πE L ) = w(E L ) = 1. Let S be the set consisting
of all archimedean places of K and all places of bad reduction of E and E L . For any ω ∈ Ψsp (S)
the corresponding quadratic extension Kω /K is unramified at all places in S and split at the nonarchimedean places of S, so that
(1/2, πE ⊗ ω) = w(E Kω ) = w(E) = −1
and similarly (1/2, πE L ⊗ ω) = w(E L ) = 1. The former implies that the choice M = Kω satisfies (a),
while the latter along with Theorem 3.2 gives that for infinitely many such ω we have that M = Kω
also satisfies (b), because
L(1, E Kω , χ) = L(1/2, πE ⊗ ηχ ⊗ ω) = L(1/2, πE L ⊗ ω).
Case p ≥ 3. The representation πE ⊗ ηχ is not self-contragredient. In fact, the set of ordinary
primes of E has density 1/2 (CM case, by Deuring) or 1 (non-CM case, by Serre), while the set of
primes p of K with χ(Frobp ) 6= 1 has density (p − 1)/p ≥ 2/3 by Chebotarev’s theorem. Hence there
is some prime p of K at which the local factors of L(s, E, χ) and L(s, E, χ−1 ) are distinct.
HILBERT’S TENTH PROBLEM FOR RINGS OF INTEGERS
9
Let S be the set consisting of the archimedean places of K and the places of bad reduction for E.
Then for each ω ∈ Ψsp (S) we have (1/2, πE ⊗ ω) = w(E Kω ) = w(E) = −1 as before, so that (a) is
satisfied for any such M = Kω . By Theorem 3.1 there are infinitely many ω ∈ Ψsp (S) such that
L(1, E Kω , χ) = L(1/2, πE ⊗ ηχ ⊗ ω) 6= 0.
Hence (b) is also satisfied if we let M = Kω for any of these infinitely many ω. This proves the claim
regarding the existence of infinitely many quadratic extensions M/K satisfying (a) and (b).
Take any of these extensions M/K. By (iii) and (a) we get that dim V (E M , L)1 is odd, and by (iv)
and (b) we get that dim V (E M , L)χ = 0. By Lemma 6.1 we see that for each non-trivial character ψ
of G one has dim V (E M , L)ψ = 0, and therefore
rk E M (L) = dim V (E M , L) = dim V (E M , L)1 = rk E M (K),
which is an odd number. This proves the result.
Now we deduce the theorem stated in the introduction.
Proof of Theorem 1.2. Let L/k be a cyclic extension of number fields of prime degree. By lemmas 5.1
and 5.2, there is an elliptic curve E/k satisfying condition (i) in Theorem 6.2. Condition (ii) holds
under the automorphy conjecture, and conditions (ii) and (iii) are part of our assumptions. Hence,
under the assumptions of Theorem 1.2, from Theorem 6.2 we get a suitable quadratic twist E M of E
such that
rk E M (L) = rk E M (k) > 0.
By Theorem 2.5 this implies that L/k is integrally Diophantine. As this holds for every such an
extension L/k, Proposition 2.3 gives that every Galois extension F/Q is integrally Diophantine. Given
an arbitrary number field K, we apply this to F the normal closure of K, hence K/Q is integrally
Diophantine by Proposition 2.2. In particular, H10(OK ) is undecidable.
Actually, the previous argument gives a slightly stronger result which we record here for the convenience of the reader:
Theorem 6.3. Let L/K be a Galois extension of number fields. Suppose that for every intermediate
field L ⊇ F ⊇ K properly contained in L, the following conjectures hold for all elliptic curves E defined
over F :
• the automorphy conjecture,
• the parity conjecture, and
• the analytic rank zero twisted BSD conjecture, for prime order characters.
Then L/K is integrally Diophantine. In particular, if H10(K) is undecidable, then so is H10(L).
7. Cyclic extensions of totally real number fields
In this section we consider cyclic extensions of prime degree of totally real fields. In this case, some
conditions in our results can be relaxed and the resulting statements can be of arithmetic interest.
However, no new consequence for Hilbert’s tenth problem is deduced in this setting, because if K is
totally real and L/K is cyclic of prime degree p, then either L is CM (this can only happen if p = 2)
or totally real, and H10(OL ) is known to be undecidable in those cases (cf. Section 2).
Note that the part regarding vanishing order of L-functions (i.e. analytic ranks) in the following
result is unconditional.
Theorem 7.1. Let K be a totally real number field or Q, and let L/K be a cyclic extension of prime
degree p. There are infinitely many elliptic curves E/K, each having infinitely many quadratic twists
E M , such that L(s, E M ) and L(s, E M ⊗ L) are entire and such that
ords=1 L(s, E M ⊗ L) = ords=1 L(s, E M )
is odd (hence, positive).
Furthermore, if p > 2 and [K : Q] ≤ 2 then every elliptic curve E/K has infinitely quadratic twists
as before. Moreover, if K = Q and if we assume the parity conjecture for elliptic curves over Q, then
in the same cases we obtain rk E M (L) = rk E M (Q) > 0.
10
M. RAM MURTY AND HECTOR PASTEN
Proof. Let E/K be an elliptic curve satisfying condition (i) in Theorem 6.2; when p > 2 any E/K
works, and in general we know that there are infinitely many such E with distinct j-invariants. Then
(ii) holds for K = Q by the modularity theorem of Wiles [32], Taylor-Wiles [31], and Breuil-ConradDiamond-Taylor [1]; in the case of K real quadratic (ii) holds by work of Freitas, Le Hung and Siksek
[11]. In the general case of K totally real, (ii) can only fail for finitely many j-invariants, see [11].
Moreover, when K = Q we also have that (iv) holds by a result of Kato [15].
Fix χ an irreducible non-trivial character of G = Gal(L/Q). The arguments of the previous section
give infinitely many quadratic twists E M for which ords=1 L(s, E M ) is odd and L(s, E M , χ) 6= 0. The
L-functions of automorphic elliptic curves over totally real fields correspond to L-functions of Hilbert
modular forms, so by general results of Shimura [27] one sees that actually L(s, E M , ψ) 6= 0 for each
ψ irreducible non-trivial character of G (because they are Galois conjugate to χ). Hence
ords=1 L(s, E M ⊗ L) = ords=1 L(s, E M ),
which is an odd positive integer. Furthermore, since (iv) holds for K = Q, one obtains that in this case
dim V (E M , L)ψ = 0 for each irreducible non-trivial ψ, hence rk E M (L) = rk E M (Q), and the claim
regarding the parity conjecture follows.
Note that it follows that for K real quadratic or Q and for cyclic extensions L/K of odd prime
degree, every elliptic curve over K is expected to have infinitely many quadratic twists with positive
rank preserved under base change to L. We believe this to be the case whenever K is replaced by a
number field having some real place; observe that this is supported (conditionally) by Theorem 6.2
and Lemma 5.1.
For the sake of completeness of our discussion on cyclic extensions of totally real number fields, let
us record here a result of Mazur and Rubin which is implicit in their work [18]. First, one can see that
in [18], the assumption of squareness of XE [2] can be replaced by the assumption of both the 2-parity
conjecture and the parity conjecture; this was explained to us by Karl Rubin. Using the available
results on the 2-parity conjecture, this gives:
Theorem 7.2 (Mazur-Rubin). Let K be a totally real number field, and let L/K be a cyclic extension
of prime degree p. Assume that the parity conjecture holds. Then there exist elliptic curves E/K such
that rk E(L) = rk E(K) = 1.
Proof. In [18] it is proved (unconditionally) that there are elliptic curves E/K with the property that
dimF2 Sel2 (E) = 1 and rk E(L) = rk E(K) (cf. Theorem 6.2 and Corollary 7.6 in [18]). The elliptic
curves produced in the proof have trivial K-rational 2-torsion and non-integral j-invariant, hence one
has (cf. (2) in [18])
r2 (E) := corankZ2 Sel2∞ (E) ≡ dimF2 Sel2 (E) mod 2.
Since the p-parity conjecture is proved for elliptic curves over totally real fields with non-integral jinvariant [22], we deduce w(E) = (−1)r2 (E) = −1. Under the assumption of the parity conjecture, we
obtain that rk E(K) is odd. From the injectivity of the map E(K)/2E(K) → Sel2 (E) and the fact
that the 2-torsion of E(K) is trivial, we get the result.
8. Final comments
Assuming the relevant conjectures required in our work and in [18], let us briefly give a more detailed
discussion on the conditional arithmetic consequences for elliptic curves. For this, let K be a number
field, L/K a cyclic extension of prime degree p and let E be an elliptic curve defined over K.
First, we are concerned with the existence of a suitable quadratic extension F/K such that the twist
E F of E satisfies
(1)
rk E F (K) = rk E F (L) > 0
while in [18] the more precise condition
(2)
rk E F (K) = rk E F (L) = 1
is considered. Our results apply to a larger class of elliptic curves than those in [18], since our approach
does not require E(K)[2] = 0 (unlike [18]); in fact, when K has some real place our approach predicts
that every E defined over K has infinitely many quadratic twists satisfying (1), as explained in the
HILBERT’S TENTH PROBLEM FOR RINGS OF INTEGERS
11
previous section. On the other hand, our approach only shows the existence of infinitely many twists
satisfying (1), while in [18] they give a lower bound for the number of twists satisfying (2) (with
conductor up to a given bound) for the elliptic curves E/K to which their results apply.
Finally, we would like to make some remarks in the direction of unconditional arithmetic results.
From the discussion in the previous section, one can see that the main obstruction to obtaining unconditional results where (1) or (2) are satisfied (using our approach or the results of [18]), is the parity
conjecture. The current state-of-the-art on the parity conjecture is the work of Dokchitser-Dokchitser
[10], which is conditional to the finiteness of the 2-primary and 3-primary parts of the ShafarevichTate group of elliptic curves. Following a different path, we would like to suggest a possible analytic
approach to relax the assumption of the parity conjecture in our work. The question seems to be
closely related to the problem of estimating averages of central values of L-functions for automorphic
representations of GL4 (which is admittedly hard). In fact, a better understanding of the latter subject should lead to non-vanishing theorems for simultaneous twists of two L-functions for automorphic
representations of GL2 , cf. p.169 in [2]. This technical tool would give a version of our results where
the analytic counterpart of (1) is replaced by the analytic counterpart of (2). Thus, it would suffice
to assume the rank part of the BSD conjecture for analytic rank 1 instead of the parity conjecture.
This is relevant because the rank part of BSD for analytic rank ≤ 1 has been proved unconditionally
in several important cases, see for instance [33].
9. Acknowledgements
We would like to express our gratitude to Bjorn Poonen for asking a question that led us to initiate
this research project. We thank Barry Mazur and Karl Rubin for explaining us the relation of their work
with the parity conjecture and the 2-parity conjecture, and for allowing us to include here Theorem
7.2 (see above). We also thank Shou-Wu Zhang for helpful discussions regarding the existing results
on the BSD conjecture, and Alexandra Shlapentokh for valuable feedback on a previous version of this
manuscript.
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Department of Mathematics and Statistics, Queen’s University
Jeffery Hall, University ave.
Kingston, ON Canada, K7L 3N6
E-mail address: murty@mast.queensu.ca
School of Mathematics, Institute for Advanced Study
1 Einstein Drive, Princeton, NJ 08540, USA
and
Department of Mathematics, Harvard University
1 Oxford Street, Cambridge, MA 02138, USA
E-mail address: hpasten@math.harvard.edu